The main theory which describes Space-Time and from which the prediction of the Big Bang comes is called General Relativity, from Einstein. This theory has several mathematical solutions and cosmologists worked to determine the most accurate. There are a class of alternatives but they all have the property that the equations which describe this solution have a singularity at $T=0$. Furthermore when this situation is examined physically it seems that there is a high density of all the Universe's matter there and then. So it is called the Big Bang.

The Singularity means that some terms become infinite and others unhelpfully become zero. So General Relativity has not been able to predict (or retrodict) what happens before, or how this process really began. The general assumption has been that it was some kind of giant Quantum Event. This assumption, when explained using a more complete theory of Quantum Gravity, may yet be correct.

However in the last few years, several mathematical cosmologists have taken seriously the idea that there was a Pre-Big Bang. Part of the reason for this may be because of the Cosmic Background Radiation data from satellites like WMAP. This data shows larger scale structure in the early universe than the older theories would have predicted.

In particular Roger Penrose has developed a view that the period since the Big Bang should be called an aeon, and that there were earlier aeons each infinitely long. This makes the Big Bang a kind of transition period between two aeons. The theory is speculative in several respects, but it is based on some mathematical constructions in General Relativity. This theory is called Conformal Cyclic Cosmology (CCC for short).

A recent short paper Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity gives the general idea. Although it is technical in places it demonstrates the kind of evidence that is motivating this theory. There are references in that paper to a book and other papers which describe that theory.

There are other theories around too, which suggest a pre-Big Bang model, perhaps other answers will mention those.

It is true that FRW metric, if redirected backwards in time, predicts a singularity.
However, when the universe's size is comparable to the Planck scale, no one really knows what truly happens.

To date there is no successful and consistent theory of quantum gravity, although a lot of partially successful ones exist.

String theory, for instance has become a leading competitor in the race for QG.

As for the expansion rate, that is the subject of Inflationary Cosmology.
The FRW metric, along with Einstein's Field Equations, yield equations of motion for the cosmos.
$$
H^2=\frac{8\pi G \rho(t)}{3}
$$

These yield different solutions depending on different mixtures of energy density constituents, if the universe is matter-dominated, an expansion rate is given, and if the universe is radiation-dominated a different one is yielded.

A truly exponential growth factor is yielded if the universe is "Dark Energy" dominated, meaning an energy density which is constant in time, regardless of the expansion of the universe.
$$
H^2=\frac{8\pi G \rho_0}{3}\Rightarrow \frac{\dot{a}}{a}=\sqrt{\frac{8\pi G \rho_0}{3}}
$$
Where $\rho_0$ is a constant.
One solution for DE was vacuum energy, but that description has it's own set of problems.
(say about 120 orders of magnitude of discrepancy...)

By the way, evidence suggests that the universe is right now expanding exponentially albeit very gradually.

Having said that, the universe might or might not be infinite per-say,
it is enough that it spans a space that is larger than our event horizon.

There is also a finite amount of energy at every instant of time in the universe,
even though, the existence of dark energy suggests that energy is added to the universe all the time.

A really nice book to "set you straight" is S. Dodelson's Introduction to Cosmology.
You only need to read the first chapter, and MAYBE chapter 8 (I think) that deals with inflation.

## Best Answer

In the context of FRW cosmology, there is no difference in the rate of time between the epochs of the evolution of the universe. You can see that from the form of the line element

$$ds^2=-dt^2+a(t)^2\gamma_{ij}dx^idx^j.$$

That is a result of the symmetries that you assume for the matter distribution (homogeneous, isotropic) and the choice of observers that you make. So the observers that follow the expansion of the Universe, which are the galaxies more or less, perceive the same time wherever and whenever they are. The cosmological time is the proper time of all the comoving observers, as it is evident from the line element.

In the case of a Schwarzschild metric and static observers

$$ds^2=-(1-\frac{2M}{r})dt^2+(1-\frac{2M}{r})^{-1}dr^2+r^2d\Omega^2,$$

it is the factor in front of dt that makes the difference and you have different time rates for observers at different positions.

There is one more point. Someone mentions the redshift and the perceived difference of the rate of time for faraway objects. That would appear to contradict what I am saying, but it isn't. The redshift effect is an observer symmetric effect. Like in the case of SR where you have two inertial observers with different velocities and each of them thinks that the others time runs slower, when both of them actually experience proper time. That is very different from the case of the static observers near a gravitating object, where there is no such symmetry. The clock of the observer that is at bigger r runs faster than the clock of the one that is at smaller r.